Band edge localization beyond regular Floquet eigenvalues

Abstract

We prove that localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in $L^2({\mathbb{R}}^d)$ is universal. By this we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations.

arXiv comment13 pages; now the stronger dynamical localization in the short range case is formulated, a few references have beed added, minor editorial changes

References in corpus (16)

• Moment Analysis for Localization in Random Schroedinger Operators
• A comprehensive proof of localization for continuous Anderson models with singular random potentials
• Bounds on the density of states for Schr\" odinger operators
• Localization for the random displacement model
• Scale-free unique continuation estimates and applications to random Schrödinger operators
• Localization for random perturbations of periodic Schroedinger operators with regular Floquet eigenvalues
• Scale-free unique continuation principle, eigenvalue lifting and Wegner estimates for random Schrödinger operators
• Characterization of the Anderson metal-insulator transition for non ergodic operators and application
• Ergodicity and dynamical localization for Delone-Anderson operators
• Conditional Wegner estimate for the standard random breather potential
• Wegner estimate and disorder dependence for alloy-type Hamiltonians with bounded magnetic potential
• Wegner estimate and localization for alloy-type models with sign-changing exponentially decaying single-site potentials
• Perturbations of continuum random Schrödinger operators with applications to Anderson orthogonality and the spectral shift function
• Lifshitz tails for Schrödinger operators with random breather potential
• A bound on the averaged spectral shift function and a lower bound on the density of states for random Schrödinger operators on ${\mathbb{R}}^d$
• A lower Wegner estimate and bounds on the spectral shift function for continuum random Schrödinger operators

Cited by in corpus (5)

• An abstract Logvinenko-Sereda type theorem for spectral subspaces
• Localisation for Delone operators via Bernoulli randomisation
• Wegner estimate for random divergence-type operators monotone in the randomness
• Quantitative unique continuation for spectral subspaces of Schrödinger operators with singular potentials
• Unique continuation for the gradient of eigenfunctions and Wegner estimates for random divergence-type operators